Logica's thoughs on numbers

[Logica]

Often I speak about how it is impossible ever to know anything as a true and unassailable fact, that assumptions are required in all things (even the scientific method, which attempts to be as objective as possible, operates on the unknowable assumption that observations of the material world can inform of its workings).

 

Invariably, I am rejoined with: "But we know 2+2=4! That is irrefutable and unassailable, and requires no assumptions."

 

Kyrisch I think you are on the right track - ALL things have a potential to be True, False or Neutral simultaneously.

 

 

2 + 2 = 4

But divide 4 by 3 and you get 1.3333333 remaining not 1

 

Divide 3 by 2 and you get renmaining 1.5 not 1

 

Note how the remainder of these numbers are > 1

 

Haven't got time to discuss now but 1 is never actually 1 exactly so 2 + 2 = NOT EXACTLY 4 (TRUE)

[Bignose]

But divide 4 by 3 and you get 1.3333333 remaining not 1

 

What do you mean by "remaining not"?

 

I don't get at all what you are saying.

 

4/3 = 1.333333 repeating because 0.3333 repeating is the decimal representation of 1/3. That's all.

[Kyrisch]

Kyrisch I think you are on the right track - ALL things have a potential to be True, False or Neutral simultaneously.

 

 

2 + 2 = 4

But divide 4 by 3 and you get 1.3333333 remaining not 1

 

Divide 3 by 2 and you get renmaining 1.5 not 1

 

Note how the remainder of these numbers are > 1

 

Haven't got time to discuss now but 1 is never actually 1 exactly so 2 + 2 = NOT EXACTLY 4 (TRUE)

 

I think you're trying to get at [math]\lim_{n\to\infty} \frac{n+1}{n}[/math]. Unfortunately, you're wrong -- the value of that limit is in fact identically one, in a similar way that [math]\sum_{n=0}^{-\infty} (0.9)\cdot 10^n = 0.999... = 1[/math].

[Logica]

What do you mean by "remaining not"?

 

I don't get at all what you are saying.

 

4/3 = 1.333333 repeating because 0.3333 repeating is the decimal representation of 1/3. That's all.

 

Between 0 and 1 there are 10 (ten) point ones (.1)

 

.1 * 10 = 1

 

In the answer 1.33333 the .333333 also tells us something about the numerical distance relationship between 0 and 1. But where do these .1's begin at O or 1?

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I think you're trying to get at [math]\lim_{n\to\infty} \frac{n+1}{n}[/math]. Unfortunately, you're wrong -- the value of that limit is in fact identically one, in a similar way that [math]\sum_{n=0}^{-\infty} (0.9)\cdot 10^n = 0.999... = 1[/math].

 

No I don't subscribe to

 

[math]\lim_{n\to\infty} \frac{n+1}{n}[/math]

Edited April 30, 2009 by Logica

[Bignose]

Between 0 and 1 there are 10 (ten) point ones (.1)

 

.1 * 10 = 1

 

 

ok

 

In the answer 1.33333 the .333333 also tells us something about the numerical distance relationship between 0 and 1. But where do these .1's begin at O or 1?

 

 

But what does this even mean? Your question doesn't make sense to me... "But where do these .1's begin at O or 1?" Where does a 0.1 begin at anywhere?

 

0.333333... does tell about the distance between 0 and 1 -- it is 1/3 of the distance between 0 and 1. Ok, so what? It is a number just like any other number. What exactly is wrong with it?

[Logica]

 

 

But what does this even mean? Your question doesn't make sense to me... "But where do these .1's begin at O or 1?" Where does a 0.1 begin at anywhere?

 

0.333333... does tell about the distance between 0 and 1 -- it is 1/3 of the distance between 0 and 1. Ok, so what? It is a number just like any other number. What exactly is wrong with it?

 

Maths Expert! Nothing is wrong with the number. I asking if we understand the number fully.

 

When we count 1 + 1.....we assume value of 1....but where do we derive that value 1 if not by the total of 0.1 * 10. The 0.1 starts at ZERO.

 

Go and think about this a bit more before you say you don't understand.

Edited April 30, 2009 by Logica

[Bignose]

Maths Expert! Nothing is wrong with the number. I asking if we understand the number fully.

 

When we count 1 + 1.....we assume value of 1....but where do we derive that value 1 if not by the total of 0.1 * 10. The 0.1 starts at ZERO.

 

Go and think about this a bit more before you say you don't understand.

 

 

No, I'm going to say I still don't understand because you aren't making yourself clear. Anyone else is free to jump in here and try to make it clearer, but I still don't understand your point. Maybe if you came right out and said exactly what your point is instead of trying to dance around the issue, it would be clearer.

 

Sure, 0.1*10 = 1

 

But, 3*0.3333.... = 1 as well. And you can do this in an infinite number of ways.

 

So, again... what is the point?

 

You never did clarify by what you meant when you wrote "But divide 4 by 3 and you get 1.3333333 remaining not 1"

 

You are using words I am familiar with, but not in a familiar way. I am simply asking you to clarify exactly what you mean when you use words in a confusing way.

 

Similarly, I don't know what you mean by "The 0.1 starts at ZERO." There is no start or end to a single number. It only is a number. If you want a start and end, you have to define a range like [math]x \in (0,0.1)[/math], then I think you can say x starts at 0. But a single number, like 0.1, doesn't start or end anywhere (unless you want to say it starts and ends at 0.1, because that is the only value)!

[Logica]

Maybe if you came right out and said exactly what your point is instead of trying to dance around the issue, it would be clearer.

 

Sure, 0.1*10 = 1

 

But, 3*0.3333.... = 1 as well. And you can do this in an infinite number of ways.

 

So, again... what is the point?

 

The point is "the point".

 

I will outline answers to your questions in about an hour cause I'm in the middle of something else at the moment. But 3*.3333 = .9999 not 1...if you can be patient calculate how many 0.1's would be needed to make up the deficit of .0001 to round off to 1.

[Bignose]

But 3*.3333 = .9999 not 1..

 

I had a fear that this was the real gist of these posts.

 

0.9999... infinitely repeating 9's = 1. They are the same number.

 

There are several proofs.

 

Let x = 0.999999999999... (where the ... after indicates infinitely repeating).

Therefore 10*x = 9.9999999999...

 

10x-x = 9x = 9.99999... - 0.999.... = 9.000000000000...

 

Divide by 9 and x = 1.00000....

 

QED

 

Another one: If 0.99999... and 1 are not the same number, then by definition if [math]a /ne b[/math] then there is some number c that is between a and b. That is there exists c such that [math]a > c > b[/math] or [math]b > c > a[/math] depending on whether a>b or b>a. So, if a=0.99999... and b = 1, and [math]a /ne b[/math], then what is c? Conversely, if no such c exists such that [math]a > c > b[/math] or [math]b > c > a[/math], then there is not other conclusion than a=b.

 

So, 1) what is the hole in the first proof, and 2) what is a number c such that 0.999... < c < 1?

 

If 0.999... and 1 really aren't the same number, then these should be easy questions to answer.

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if you can be patient calculate how many 0.1's would be needed to make up the deficit of .0001 to round off to 1.

 

Again, I am sorry, but I don't know what this means. How do you calculate "how many 0.1's would be needed to make up the deficit of .0001"? What deficit? What do you mean by deficit?

 

How many makes up I read as division: 0.001/0.1 = 0.0001 --> which is how many 0.1's "make up" 0.001. But, I don't that this is what you are looking for.

 

And rounding is an entirely different mathematical operation.

 

So, once again, you are using words I am familiar with in very unfamiliar ways. When you write about science and math, you need to be very exact in the word choices you make, because each word has a very specific definition associated with it. This is different than regular English where words have different meanings -- in math, words only have one meaning so that everyone knows exactly what is meant when that word is said. Mathematically, when I write "multiply" everyone knows that I mean the multiplication operator and not the exponentiation operator, even though they are related. Whereas when writing a novel or story, you can be loose with terms like multiplication or rounding or deficit -- when you write math, you need to be very precise in what you mean. So, I ask, please be more precise, more exact in what you mean when you write these things so that I and everyone else familiar with the mathematics will understand you and your points better.

Edited May 1, 2009 by Bignose
Consecutive posts merged.

[Logica]

 

1) what is the hole in the first proof, and 2) what is a number c such that 0.999... < c < 1?

 

If 0.999... and 1 really aren't the same number, then these should be easy questions to answer.

 

I posted in this thread because the OP writes:

 

In conclusion, it seems to me that the whole body of math does not include any actual knowledge, just an infrastructure of definition based entirely on the abstract idea of quantity.

 

The gist of ALL my posts here was an attempt to challenge assumptions...not just some covert way to refute 0.9999... infinitely repeating 9's = 1. I am interested in O and questions O defined etc.

 

 

1) What is the hole in the first proof.

 

The use of the function y=1-1/x is a mathematical construct to compensate for the fact that .999... cannot achieve 1. It is numerically impossible because computation is locked in at .999... and although the function appears to resolve .999... = 1 it is at the same time fudging this result.

 

 

 

So my proof:

 

Let x = 0.1

Therefore 10*x = 1

x = .1

 

 

.1 X .09999 = .999... + y = 1

 

y is indeterminate in this case.

 

 

 

2) What is a number c such that 0.999... < c < 1?

 

Note .9 and .1 positions are equidistant in relation to 1

both are separated from 1 by .1

 

But because .999... can never reach 1 we must concede that .1 is closer to 1 than .999... and thus .999...cannot = 1

 

.999... < .1 < 1

 

Please do not lecture me - I am not a mathematician but I do not think this should preclude me from participating in this thread listed under Speculation & Pseudoscience thread.

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I see you are online now but it is 3:10 am and I cannot hang around for your response. Good night!

[Kyrisch]

There are multiple things wrong with this post. The most glaring error is that 0.999... is greater than 0.1, irrefutably. Further, 0.999... isn't "going" anywhere, it is a single number with a single value. The reason that you cannot reconcile with your mathematical sense that 0.999... = 1 is because you cannot conceive of infinity.

[Bignose]

 

.999... < .1 < 1

 

 

This is dead wrong. 0.99999... is less than 0.1? In what universe?

 

 

The use of the function y=1-1/x is a

 

where did I post the use of the function 1-1/x? This has nothing to do with what I wrote....

 

 

 

So my proof:

 

Let x = 0.1

Therefore 10*x = 1

x = .1

 

.1 X .09999 = .999... + y = 1

 

y is indeterminate in this case.

 

 

This isn't any proof at all! You can't just throw in a y and call it "indeterminate" and call that a proof.

 

Besides that, there is a pretty obvious error here, too. 0.1 x 0.099........ would be equal to 0.00999....

 

Please do not lecture me - I am not a mathematician but I do not think this should preclude me from participating in this thread listed under Speculation & Pseudoscience thread.

 

Finally, I didn't lecture you. I tried to make an appeal for you to use precise language so that everyone understood exactly what you mean. I am trying to understand your posts so that I can respond intelligently. You've never answered some of my questions, and you've made some pretty basic errors in those you did answer (0.99999 less than 0.1 !!!), so I guess it wasn't worth my effort.

 

You don't have the be a mathematician to use the words correctly. I'm not a mathematician, but I know what words I use I use correctly. If you aren't aware of their specific meanings, there is no shame in that -- just look up the precise meanings or ask and this forum will help. I'll help, I promise.

 

Look, it's no different than when you first learn a game, you learn the terminology used in that game. Take baseball for example. If you started using your own words for balls and strikes for example, no one would know what you are talking about. "The count is 1 banana and 2 envelopes." Or if you used different terms for safe and out. "That runner was clearly polka dotted!" Or, take chess for example. You can't make a move like "Duke to Cardinal's 23rd, check". It doesn't mean anything in the accepted terminology of the game.

 

Math is the same way. There are terms that have been accepted and are used in one way and only one way. Again, if you don't know them currently, then it only takes a tiny bit to learn how to use them right and then use correct language -- everyone benefits because then there won't be several posts asking you to clarify what you write because you are using terms that aren't defined.

 

I don't mean this to sound like a lecture -- It is just an appeal to use terms properly so that everyone can understand what you are trying to say. Because, I still don't know what most of your first posts in this thread were trying to say.

[Klaynos]

It is important that members answer questions that are put to them directly about their ideas.

 

It is also important that discussions are kept free of personal insults, no matter how subtle.

 

Heed this warning, failure to do so will result in infractions.